We can't use this as an example, Carnegie did not become wealthy in an anarchic society. All anarchists would agree that concepts of ownership can only apply after the unfair distribution of wealth caused by the state has been remedied.

However.... we can construct a similar thought experiment.

I decide to build a factory, I have accumulated resources through working hard throughout much of my life and making sound business decisions. I offer to pay you some of these resources every week if you build a factory for me, in return for this, you agree that the final product belongs to me, and that you are just a builder. In this circumstance, you have been bought out of your share of the factory in advance, in effect, because you agreed to be paid now. You might say "No! I must also be allowed to work in the factory, and to buy my way in, if you pay me a bit less" (an allusion to the power of unions in an anarcho-capitalist society) - we'd need to come to an agreement, but lets just say I built the factory without giving away ownership and I have people working for me in return for wages.

In another circumstance, a group of syndicalists decide to build a factory, none of them have accumulated enough resources to do it themselves, so they decide that the factory belongs to all of them once it is built, and they pool their resources together to make it happen.

As long as in the first circumstance I accumulated those resources through my own efforts and through fair means (i.e the state did not grant me a tract of land to call my own and a title by which I could extract rent for it, or my father could etc.), and did not betray any contracts, I have done nothing wrong, you agreed that you wouldn't own the factory, and in return I'd pay you for building it, why should you be allowed to claim ownership now that its finished, just because I worked hard and saved resources in order to pay you while you didn't?

The difference is, the syndicalists had a genuine grievance because they had no fair access to resources in order to build a factory, as the apparatus of the state did not create a level playing field. Don't assume anarcho-capitalists don't recognise that inequality is forced upon us now, but we can still imagine a society after this has been adjusted.

In an anarchist society, you are responsible for accumulating and maintaining your own wealth so that you can own factories, otherwise you'll have to look at collective ownership. In all likelihood most workers wouldn't work for people who didn't allow "buy-ins", I know I certainly wouldn't, because the continuing success of a business relies on innovation that can come from anywhere, but some people would just want money at the end of the week and would be happy, there's no reason both can't co-exist among freely associating individuals.

So you syndicalists can pool your wealth together and build a factory, and the capitalist can pay some builders to build him one. As long as no coercion was involved in acquiring the capitalists wealth, both should be perfectly tolerable?